1915]
THE PRINCIPLE OF SIMILITUDE
303
where F is an arbitrary function of the one variable avc/K. An important particular case arises when the solid takes the form of a cylindrical wire of any section, the length of which is perpendicular to the stream. In strictness similarity requires that the length I be proportional to the linear dimension of the section b ; but when I is relatively very great h must become proportional to I and a under the functional symbol may be replaced by b. Thus
We see that in all cases h is proportional to 0, and that for a given fluid F is constant provided v be taken inversely as a or b.
In an important class of cases Boussinesq has shown that it is possible to go further and actually to determine the form of F. When the layer of fluid which receives heat during its passage is very thin, the flow of heat is practically in one dimension and the circumstances are the same as when the plane, boundary of a uniform conductor is suddenly raised in temperature and. so maintained. From these considerations it follows that F varies as v%, so that in the case of
the wire
h oc W. \f(bvcl/c),
the remaining constant factor being dependent upon the shape and purely numerical. But this development scarcely belongs to my present subject.
It will be remarked that since viscosity is neglected, the fluid is regarded as flowing past the surface of the solid with finite velocity, a serious departure from what happens in practice. If we include viscosity in our discussion, the question is of course complicated, but perhaps not so much as might be expected. We have merely to include another factor, vw, where v is the kinematic viscosity of dimensions (Length)'"1 (Time)"1, and we find by the same process as before
, n fQiVC\ ICv\
h = Ka0. — • —) •
\ K J U/
Here z and w are both undetermined, and the conclusion is that
h =
avc
K
GV
K
where F is an arbitrary function of the two variables avc/K and CV/K. The latter of these, being the ratio of the two diffusivities (for momentum and for temperature), is of no dimensions; it appears to be constant for a given kind of gas, and to vary only moderately from one gas to another. If we may assume the accuracy and universality of this law, cvj/c is a merely numerical constant, the same for all gases, and may be omitted, so that h reduces to the forms already given when viscosity is neglected altogether, F being again a function of a single variable, avc/tc or bvcfx. In any case F is constant for a given fluid, provided v be taken inversely as a or b.
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